Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.
Problem 2: For an underdamped system (8< wo), the two solutions can be written as x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin w₁t. (a) Show that as →→ wo, x₁(t) approaches the critically damped solution e-ßt. (b) What happens to r₂(t)? Show that the expression x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second critically damped solution te-ßt in the limit 3 → wo.
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![Problem 2: For an underdamped system (3< wo), the two solutions can be written as
x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin wit. (a) Show that as → wo, i(t) approaches
the critically damped solution e-Bt. (b) What happens to r₂(t)? Show that the expression
x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second
critically damped solution te-ßt in the limit ß→wo.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29c74d06-0f3b-4eb2-9c9d-dbbc1918002c%2F8f88dac9-2abc-403a-bf5b-09124eaaac67%2Fef703a_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2: For an underdamped system (3< wo), the two solutions can be written as
x₁(t) = e-ßt cosw₁t and x₂(t) = e-ßt sin wit. (a) Show that as → wo, i(t) approaches
the critically damped solution e-Bt. (b) What happens to r₂(t)? Show that the expression
x₂(t)/W₁ (which is still a valid solution to the underdamped system) approaches the second
critically damped solution te-ßt in the limit ß→wo.
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